The latent variable proximal point (LVPP) algorithm is a framework for solving infinite-dimensional variational problems with pointwise inequality constraints. The algorithm is a saddle point reformulation of the Bregman proximal point algorithm. Although equivalent at the continuous level, the saddle point formulation is significantly more robust after discretization.

 

LVPP yields simple-to-implement numerical methods with robust convergence and observed mesh-independence for obstacle problems, contact, fracture, plasticity, and others besides; in many cases, for the first time. The framework also extends to more complex constraints, providing means to enforce convexity in the Monge--Ampère equation and handling quasi-variational inequalities, where the underlying constraint depends implicitly on the unknown solution. Moreover the algorithm is largely discretization agnostic allowing one to discretize with very-high-order $hp$-finite element methods in an efficient manner. In this talk, we will describe the LVPP algorithm in a general form and apply it to a number problems from across mathematics.